Conidia and oidia are asexual spores. The term conidium is usually used to refer to asexual spores that are the fungus' main (or substantive) reproductive route (at least in the current anamorph). As such, they are often born on elaborate structures called conidiophores. Gonidium is an obsolete form of conidium.
Conidia are grown on elaborate structures called conidiophores. These are usually stalked, lifting the conidia off the substrate for better dispersal and to avoid microscopic grazing animals. They often produce hundreds or thousands of conidia at a time.
In contrast, oidium is usually used to refer to asexual spores that are born a few at a time on very simple hyphae that just stick out a little bit into the substrate, and are (rightly or wrongly) presumed not to constitute the main reproductive preoccupation of the fungus at that time. Many Coprinusspecies, for example, produce oidia on their mycelium, even though they're also busy pushing up sexual fruiting bodies.
Arthrospores are conidia that are produced very simply by the last cell on a hypha breaking off and dispersing as a propagule. This seems to have been Nees' conception of how his Aspergillus worked in the picture above. He was wrong, but it does make a good illustration of arthrospores.
